Let X be a topological space, A
a subset of X, and F : X → X a map. Suppose there exists a retraction ρ : W → A
where A ∪ F(A) ⊆ W ⊆ X; then define f : A → A by f(a) = ρF(a). The map f is
called a retract of F. If all the fixed points of f are fixed points of F, we say that F is
retractible onto A (with respect to ρ). Then, if A is a compact ANR, the Nielsen
number N(f) of f is a lower bound for the number of fixed points of F, or of any
map G : X → X retractible onto A with retract homotopic to f. Many classes of
examples of retractible maps can be found, even if X is required to be a euclidean
space. If F is retractible onto a compact ANR with respect to a deformation
retraction of X onto A, then we say that F is deformation retractible (dr) and we
define a number D(F) which we prove to have the property: if G : X → X is
a dr map homotopic to F, then G has at least D(F) fixed points. If X is
an ANR and F is a compact map, then D(F) is the Nielsen number of F.
We find conditions, for any map F : X → X retractible onto A, so that
there exists G : X → X retractible onto A and with retract homotopic to
f such that G has exactly N(f) fixed points. Furthermore, if F is dr, the
hypotheses yield a dr map G homotopic to F and with exactly D(F) fixed points.
These last results are based on a technique, of independent interest, for
extending a map g : A → A, on a finite subpolyhedron of a locally finite
polyhedron X, to a map G : X → X in such a way that G has no fixed points on
X − A.
